The maximum entropy principle does give implausible results if applied carelessly but the above reasoning seems very strange to me. The normal way to model this kind of scenario with the maximum entropy prior would be via Laplace’s Rule of Succession, as in Max’s comment below. We start with a prior for the probability that a randomly drawn ball is red and can then update on 99 red balls. This gives a ^{100}⁄_{101} chance that the final ball is red (about 99%!). Or am I missing your point here?

Somewhat more formally, we’re looking at a Bernoulli trial—for each ball, there’s a probability p that it’s red. We start with the maximum entropy prior for p, which is the uniform distribution on the interval [0,1] (= beta(1,1)). We update on 99 red balls, which gives a posterior for p of beta(100,1), which has mean ^{100}⁄_{101} (this is a standard result, see e.g. conjugate priors - the beta distribution is a conjugate prior for a Bernoulli likelihood).

The more common objection to the maximum entropy principle comes when we try to reparametrise. A nice but simple example is van Fraassen’s cube factory: a factory manufactures cubes up to 2x2x2 feet, what’s the probability that a randomly selected cube has side length less than 1 foot? If we apply the maximum entropy principle (MEP), we say ^{1}⁄_{2} because each cube has length between 0 and 2 and MEP implies that each length is equally likely. But we could have equivalently asked: what’s the probability that a randomly selected cube has face area less than 1 foot squared? Face area ranges from 0 to 4, so MEP implies a probability of ^{1}⁄_{4}. All and only those cubes with side length less than 1 have face area less than 1, so these are precisely the same events but MEP gave us different answers for their probabilities! We could do the same in terms of volume and get a different answer again. This inconsistency is the kind of implausible result most commonly pointed to.

Thanks for the clarification—I see your concern more clearly now. You’re right, my model does assume that all balls were coloured using the same procedure, in some sense—I’m assuming they’re independently and identically distributed.

Your case is another reasonable way to apply the maximum entropy principle and I think it’s points to another problem with the maximum entropy principle but I think I’d frame it slightly differently. I don’t think that the maximum entropy principle is actually directly problematic in the case you describe. If we assume that all balls are coloured by completely different procedures (i.e. so that the colour of one ball doesn’t tell us anything about the colours of the other balls), then seeing 99 red balls doesn’t tell us anything about the final ball. In that case, I think it’s reasonable (even required!) to have a 50% credence that it’s red and unreasonable to have a 99% credence, if your prior was 50%. If you find that result counterintuitive, then I think that’s more of a challenge to the assumption that the balls are all coloured in such a way that learning the colour of some doesn’t tell you anything about the colour of the others rather than a challenge to the maximum entropy principle. (I appreciate you want to assume nothing about the colouring processes rather than making the assumption that the balls are all coloured in such a way that learning the colour of some doesn’t tell you anything about the colour of the others, but in setting up your model this way, I think you’re assuming that implicitly.)

Perhaps another way to see this: if you don’t follow the maximum entropy principle and instead have a prior of 30% that the final ball is red and then draw 99 red balls, in your scenario, you should maintain 30% credence (if you don’t, then you’ve assumed something about the colouring process that makes the balls not independent). If you find that counterintuitive, then the issue is with the assumption that the balls are all coloured in such a way that learning the colour of some doesn’t tell you anything about the colour of the others because we haven’t used the principle of maximum entropy in that case.

I think this actually points to a different problem with the maximum entropy principle in practice: we rarely come from a position of complete ignorance (or complete ignorance besides a given mean, variance etc.), so it’s actually rarely applicable. Following the principle sometimes gives counterintuive/unreasonable results because we actually know a lot more than we realise and we lose much of that information when we apply the maximum entropy principle.